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Annegret, Construction of CM Picard curves
 Math. Comp
"... Abstract. In this article we generalize the CM method for elliptic and hyperelliptic curves to Picard curves. We describe the algorithm in detail and discuss the results of our implementation. 1. ..."
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Abstract. In this article we generalize the CM method for elliptic and hyperelliptic curves to Picard curves. We describe the algorithm in detail and discuss the results of our implementation. 1.
E.: Primes in the denominators of Igusa class polynomials
, 2003
"... The purpose of this note is to suggest an analogue for genus 2 curves of part of Gross and Zagier’s work on elliptic curves [GZ84]. Experimentally, for genus 2 curves with CM by a quartic CM field K, it appears that primes dividing the denominators of the discriminants of the Igusa class polynomials ..."
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The purpose of this note is to suggest an analogue for genus 2 curves of part of Gross and Zagier’s work on elliptic curves [GZ84]. Experimentally, for genus 2 curves with CM by a quartic CM field K, it appears that primes dividing the denominators of the discriminants of the Igusa class polynomials all have the property
Igusa class polynomials, embeddings of quartic CM fields, and arithmetic intersection theory, in Cojocaru et al
"... Abstract. Bruinier and Yang conjectured a formula for an intersection number on the arithmetic Hilbert modular surface, CM(K).Tm, where CM(K) is the zerocycle of points corresponding to abelian surfaces with CM by a primitive quartic CM field K, and Tm is the HirzebruchZagier divisors parameterizi ..."
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Abstract. Bruinier and Yang conjectured a formula for an intersection number on the arithmetic Hilbert modular surface, CM(K).Tm, where CM(K) is the zerocycle of points corresponding to abelian surfaces with CM by a primitive quartic CM field K, and Tm is the HirzebruchZagier divisors parameterizing products of elliptic curves with an misogeny between them. In this paper, we examine fields not covered by Yang’s proof of the conjecture. We give numerical evidence to support the conjecture and point to some interesting anomalies. We compare the conjecture to both the denominators of Igusa class polynomials and the number of solutions to the embedding problem stated by Goren and Lauter. 1.
Computing Igusa class polynomials
, 2008
"... We give an algorithm that computes the genus two class polynomials of a primitive quartic CM field K, and we give a runtime bound and a proof of correctness of this algorithm. This is the first proof of correctness and the first runtime bound of any algorithm that computes these polynomials. Our alg ..."
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We give an algorithm that computes the genus two class polynomials of a primitive quartic CM field K, and we give a runtime bound and a proof of correctness of this algorithm. This is the first proof of correctness and the first runtime bound of any algorithm that computes these polynomials. Our algorithm uses complex analysis and runs in time e O( ∆ 7/2), where ∆ is the discriminant of K. 1
Limitations of Constructive Weil Descent
"... Weil restriction of scalars can be used to construct curves suitable for cryptography whose Jacobian has known group order. The description given by Hess, Seroussi and Smart shows how to construct hyperelliptic curves of genus two or three over finite fields of characteristic two. One drawback of th ..."
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Weil restriction of scalars can be used to construct curves suitable for cryptography whose Jacobian has known group order. The description given by Hess, Seroussi and Smart shows how to construct hyperelliptic curves of genus two or three over finite fields of characteristic two. One drawback of their method is discussed, namely that only a small proportion of the set of all curves can be constructed in this way. We also indicate that the method cannot be generalised to overcome this drawback.
EXPLICIT CMTHEORY IN DIMENSION 2
"... Abstract. For a complex abelian surface A with endomorphism ring isomorphic to the maximal order in a quartic CMfield K, the Igusa invariants j1(A), j2(A), j3(A) generate an abelian extension of the reflex field of K. In this paper we give an explicit description of the Galois action of the class g ..."
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Abstract. For a complex abelian surface A with endomorphism ring isomorphic to the maximal order in a quartic CMfield K, the Igusa invariants j1(A), j2(A), j3(A) generate an abelian extension of the reflex field of K. In this paper we give an explicit description of the Galois action of the class group of this reflex field on j1(A), j2(A), j3(A). We give a geometric description which can be expressed by maps between various Siegel modular varieties. We can explicitly compute this action for ideals of small norm, and this allows us to improve the CRT method for computing Igusa class polynomials. Furthermore, we find cycles in isogeny graphs for abelian surfaces, thereby implying that the ‘isogeny volcano ’ algorithm to compute endomorphism rings of ordinary elliptic curves over finite fields does not have a straightforward generalization to computing endomorphism rings of abelian surfaces over finite fields. 1.
Computing genus 2 curves from invariants on the Hilbert moduli space
 J. Number Theory
"... Abstract. We give a new method for generating genus 2 curves over a finite field with a given number of points on the Jacobian of the curve. We define two new invariants for genus 2 curves as values of modular functions on the Hilbert moduli space and show how to compute them. We relate them to the ..."
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Abstract. We give a new method for generating genus 2 curves over a finite field with a given number of points on the Jacobian of the curve. We define two new invariants for genus 2 curves as values of modular functions on the Hilbert moduli space and show how to compute them. We relate them to the usual three Igusa invariants on the Siegel moduli space and give an algorithm to construct curves using these new invariants. Our approach simplifies the complex analytic method for computing genus 2 curves for cryptography and reduces the amount of computation required. 1.
PAIRINGS ON HYPERELLIPTIC CURVES
, 2009
"... We assemble and reorganize the recent work in the area of hyperelliptic pairings: We survey the research on constructing hyperelliptic curves suitable for pairingbased cryptography. We also showcase the hyperelliptic pairings proposed to date, and develop a unifying framework. We discuss the techni ..."
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We assemble and reorganize the recent work in the area of hyperelliptic pairings: We survey the research on constructing hyperelliptic curves suitable for pairingbased cryptography. We also showcase the hyperelliptic pairings proposed to date, and develop a unifying framework. We discuss the techniques used to optimize the pairing computation on hyperelliptic curves, and present many directions for further research.
NONCYCLIC SUBGROUPS OF JACOBIANS OF GENUS TWO CURVES WITH COMPLEX MULTIPLICATION
"... Abstract. Let E be an elliptic curve de ned over a nite eld. Balasubramanian and Koblitz have proved that if the ℓth roots of unity µℓ is not contained in the ground eld, then a eld extension of the ground eld contains µℓ if and only if the ℓtorsion points of E are rational over the same eld extens ..."
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Abstract. Let E be an elliptic curve de ned over a nite eld. Balasubramanian and Koblitz have proved that if the ℓth roots of unity µℓ is not contained in the ground eld, then a eld extension of the ground eld contains µℓ if and only if the ℓtorsion points of E are rational over the same eld extension. We generalize this result to Jacobians of genus two curves with complex multiplication. In particular, we show that the Weil and the Tatepairing on such a Jacobian are nondegenerate over the same eld extension of the ground eld. 1.
PAIRINGS ON JACOBIANS OF HYPERELLIPTIC CURVES
"... Abstract. Consider the Jacobian of a hyperelliptic genus two curve de ned over a nite eld. Under certain restrictions on the endomorphism ring of the Jacobian, we give an explicit description of all nondegenerate, bilinear, antisymmetric and Galoisinvariant pairings on the Jacobian. From this des ..."
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Abstract. Consider the Jacobian of a hyperelliptic genus two curve de ned over a nite eld. Under certain restrictions on the endomorphism ring of the Jacobian, we give an explicit description of all nondegenerate, bilinear, antisymmetric and Galoisinvariant pairings on the Jacobian. From this description it follows that no such pairing can be computed more e ciently than the Weil pairing. To establish this result, we need an explicit description of the representation of the Frobenius endomorphism on the ℓtorsion subgroup of the Jacobian. This description is given. In particular, we show that if the characteristic polynomial of the Frobenius endomorphism splits into linear factors modulo ℓ, then the Frobenius is diagonalizable. Finally, under the restriction that the Frobenius element is an element of a certain subring of the endomorphism ring, we prove that if the characteristic polynomial of the Frobenius endomorphism splits into linear factors modulo ℓ, then the embedding degree and the total embedding degree of the Jacobian with respect to ℓ are the same number. 1.